#StackBounty: #statistical-significance #joint-distribution Significant Test for Joint Probability Distributions

Bounty: 50

I am familiar with significant tests and distance measure between standard probability distributions. However, I am looking for a significance test between joint probability distributions (JPD) for a set of ecological data. (The data is discrete and categorical).

As an example, we have the following two joint probability distributions, ${P_1}$ and ${P_2}$. (Categories in rows; attributes in columns).

$${P_1}=
begin{pmatrix}
0.203 & 0.203 & 0.020 \
0.033 & 0.229 & 0.033 \
0.059 & 0.072 & 0.150 \
end{pmatrix}
$$

$${P_2}=
begin{pmatrix}
0.159 & 0.051 & 0.025 \
0.080 & 0.239 & 0.051 \
0.040 & 0.188 & 0.167 \
end{pmatrix}
$$

with the respective sample sizes being ${N_{P_1}=153}$; ${N_{P_2}=276}$

The respective, if not obvious, hypotheses are: ${{H_0}: {P_1}={P_2}}$ and ${{H_A}: {P_1}neq{P_2}}$.

Is there a significance test for this situation? My assumption there is such a test (after all, there seems to be a test for everything) and that I have simply not come across yet. Any pointers would be most welcome.

(Note: I have already done a range of other tests (e.g. Chi-squared test, entropy) for each JPD. I have also calculated the Hellinger distance between each JPD. This question relates explicitly to a significance test between two JPD).


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